is? •<

II f

156 Practical Mathematics

followed very closely hy the first five observations, hut that, as the weight increases,
the same law is not so accurately followed.

EXAMPLE ($}.—Newton's law of cooling. The following- are the results en m
m&nn's experiments to find the law which governs the rate of cooling of a heated vt
suspended in air. & is the excess in temperature of the body over the temperature of
surroundings, at time t seconds from the beginning of the experiment.

®
19-9
18-9
16-9
14-9
12-9
10*9
8-9
6'9

i
0
3'4S
10-85
19-30
28-80
40' 10
5375
7°'95

According to Newton's law of cooling we should have

Q =

where a is constant and 0{ = the temperature when, t is o = 19*9.

To test how far the above results follow this law, we have, taking logs of 9 —

log; 9
I '2989
1-2765
T2279
ri732
riio6
1-0374
0-9494
0-8388

Plotting log B and t we get a straight line sloping downwards as / increases, thus
verifying that 9 and t follow a law of the form 9 = Q^trat.
Taking logs, we get

loglft 0 = - at Iogl9 f -h logu 0i = - 0-43430 . t -f 10&. 0,

as the equation to this straight line.

Substituting the co-ordinates of two points on the line, and solving the resulting
simultaneous equations, we get

tf = 0*015, 0t = 19*9

„*. the temperature and the time, are connected by the law 9 = ig'ge-0*1**.

The student should draw the figure from the above data, and verify this result.

EXAMPLES.—LIL

Find the law connecting^ and x in the following cases :•—
1.

X
o*4
O72
„
i-5
2

y
3-3*
87
*7-3
9i
407

%

X
2-4
3'6
4-8
5'3
6-9

y
«6
«.
375
4*9
107

1